Problem in computing energy functional with two memories

Question

We consider the system:$$\eqalign{ & {u_t}(t,x) = {u_{xx}}(t,x) + \int\limits_0^t {{g_1}(t-s){v_{xx}}(s,x)} ds \cr & {v_t}(t,x) = {v_{xx}}(t,x) + \int\limits_0^t {{g_2}(t-s){u_{xx}}(s,x)} ds \cr & u(t,0) = u(t,1) = v(t,0) = v(t,1) = 0 \cr} $$ The usual method to compute the energy functional is to multiply the first equation by ${u_t}$ and the second by ${v_t}$ and integrating by part, I found a problem in the two memory terms $$\int\limits_0^t {{g_1}(t - s)\int\limits_0^L {{v_{xx}}(s,x)} } u(t,x)dxds$$ and $$\int\limits_0^t {{g_2}(t - s)\int\limits_0^L {{u_{xx}}(s,x)} } v(t,x)dxds.$$ How can I deal with two terms and write them on the form $${1 \over 2}{d \over {dt}}\{ ...\}.$$